How to calculate the cross-section area of an object?
One method is:


Enable the "3D Print Toolbox" (follow instructions here).
It can make it easier to move the object to be measured to the origin such that the axis for which you want the cross-section is aligned with a coordinate axis.
Create a cube.
Move the cube using the text entry feature to align with the axis of the object for which you want the cross-section.
For the axis perpendicular to the cross-section, scale this side of the cube to almost nothing (e.g. 0.000001 should be enough).
Scale the other two sides of the cube such that they are larger than the object cross-section to be measured.
Add a Boolean modifier to the cube with an Intersect Operation on the object. Apply the modifier.
Select the object and press H to hide it, so the cross-section profile can be selected.
Select the cross-section profile. Select 3D view > Tool shelf > 3D Printing tab, and select the Area button.
Divide the displayed area result by two (the area measured is the whole surface area of the intersecting object. Ignoring the insignificant side area, that leaves the front and back areas contributing to the total cross-section area).


This illustrates the process visually (unfortunately some recorded key presses were accidental):



Useful resource: Create Cross Sections and 2D planes in Blender